1. **State the problem:** We are given a triangle with points B, K, C, L, and D. Segments BK = 10, KC = 5, BL = 13, and LD = y. We need to find the length of segment LD, denoted as $y$. 2. **Analyze the figure:** Points K and L lie on a diagonal and horizontal line respectively. Since BL and LD are horizontal segments extending from B to L and L to D, and BK and KC form parts of the triangle, we can use the properties of similar triangles or the Pythagorean theorem if right angles are involved. 3. **Assumption:** Since BL = 13 and BK = 10, and KC = 5, the total length BC = BK + KC = 15. 4. **Using the triangle properties:** If BL and LD lie on the same horizontal line, then BD = BL + LD = 13 + y. 5. **Using similarity or Pythagorean theorem:** Since BK and BL share point B, and K and L lie on a diagonal, we can consider triangle BKC and triangle BLD. 6. **Calculate y:** Since BK = 10, BL = 13, and KC = 5, the ratio of BL to BK is $\frac{13}{10} = 1.3$. Assuming the triangles are similar, the segment LD corresponds to KC scaled by the same ratio. 7. **Compute y:** $$y = KC \times \frac{BL}{BK} = 5 \times 1.3 = 6.5$$ 8. **Final answer:** $$\boxed{6.5}$$